A regularized product-kernel transform is rebuilt so that lithium-ion observers, thermally limited power split, delayed multi-echelon inventory, and unit-load reallocation occupy one testing space. Energy-throughput identities taken from competition-pack sizing fix the admissible current. Incomplete carrier telemetry and swipe-access events sit in the same dual as demand shocks. Integer-order Thevenin maps and Caputo diffusion multipliers become the symbols inverted on a Bromwich contour. Smart-mobility and solar-assisted profiles change only the exogenous current. Spectral residues isolate thermal poles that reappear as inventory restoration modes. Green and provenance functionals attach carbon and ledger coordinates to the inverted image. Digital-twin resilience metrics become contour integrals. A reduced coefficient vector after Gaussian smoothing feeds an annealing layer for unit-load configuration. Time-shift and sequential-convergence identities keep calendar lags exact. Pack estimation, control-tower twins, and combinatorial recovery are therefore views of one operator.

Keywords: operational calculus; lithium-ion observers; supply-chain resilience; digital twins; quantum annealing; product-kernel time-shift; green logistics.